Finite Group Databasebeta

← #374791

Subgroups — #374791

Order 512.

1198 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1990C211yesmaximal, characteristic
2C2×#1988C211yesmaximal, characteristic
3C2×C8×C4⋊C4C211yesmaximal, characteristic
4#14155C211yesmaximal, characteristic
5#14151C211yesmaximal, characteristic
6#11315C212yesmaximal
7#11292C212yesmaximal
8#10880C212yesmaximal
9#10849C211yesmaximal, characteristic
10#10820C212yesmaximal
11C4×#340C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×C4×C8C2212yes—
15C22×C4×C8C2211yescharacteristic
16C2×#391—21no—
17C2×#391—21no—
18C2×#389—21no—
19C2×#389C2212yes—
20C2×#384C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24#1990C418yescomplemented
25#1988—24no—
26C8×C4⋊C4—24no—
27C2×#340C2211yescharacteristic
28C2×#340C2212yes—
29C2×#340C2211yescharacteristic
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2211yescharacteristic
37C2×#336C2212yes—
38#1445—22no—
39#1441—22no—
40C4⋊#95—21no—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44C23×Q8—21no—
45C23×C8C2311yescharacteristic
46C23×C8C2312yes—
47C23×C8C2311yescharacteristic
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8C2×C414yes—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C42—21no—
67C22×C42—21no—
68C22×C42C2312yes—
69C22×C42C2311yescharacteristic
70C2×C4⋊C8—22no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8D414yes—
73C2×C4×C8—22no—
74C2×C4×C8—22no—
75C2×C4×C8—24no—
76C2×C4×C8—24no—
77C2×C4×C8—42no—
78C2×C4×C8C2×C414yes—
79#391—44no—
80#391—44no—
81#389—24no—
82#389—42no—
83#389—44no—
84#384—22no—
85#384C2×C414yes—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—22no—
101C2×#95D412yes—
102#340—24no—
103#340—24no—
104#340—22no—
105#340—22no—
106#340—24no—
107#340—22no—
108#340—42no—
109#340C2×C418yes—
110#340C2×C418yes—
111#340Q814yescomplemented
112#336—24no—
113C22×Q8—24no—
114C22×Q8—42no—
115C22×Q8—42no—
116C22×Q8—42no—
117C22×Q8—42no—
118C22×Q8—42no—
119C22×Q8—42no—
120C23×C4—22no—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4C2×D412yes—
127C23×C4C2×D412yes—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2411yescharacteristic
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—22no—
139C22×C8—21no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—22no—
147C22×C8C22×C412yes—
148C22×C8C22×C412yes—
149C22×C8C2×D414yes—
150C22×C8C2×Q812yes—
151C22×C8#4112yes—
152C4⋊Q8—28no—
153C4⋊Q8—28no—
154C4⋊Q8—44no—
155C4⋊Q8—44no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—81no—
200C2×C4⋊C4—81no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4D814yes—
203C2×C4⋊C4SD1614yes—
204C2×C4⋊C4C22×C414yes—
205C2×C4⋊C4C22×C414yes—
206C2×C4⋊C4C2×Q812yes—
207C2×C4⋊C4C2×Q812yes—
208C2×C4⋊C4#4112yes—
209C2×C4⋊C4#4112yes—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—42no—
217C2×C42—42no—
218C2×C42—42no—
219C2×C42—42no—
220C2×C42—42no—
221C2×C42—42no—
222C2×C42C22×C412yes—
223C2×C42C2×D412yes—
224C2×C42#4112yes—
225C2×C42#4112yes—
226C2×C42#4112yes—
227C2×C42#4112yes—
228C4⋊C8—28no—
229C4×C8—28no—
230C4×C8—48no—
231#95—24no—
232#95—44no—
233#95—42no—
234#95—42no—
235#95—44no—
236#95—44no—
237#95—44no—
238#95—44no—
239#95—44no—
240#95—82no—
241C24C22×D411yescharacteristic
242C2×Q8—48no—
243C2×Q8—48no—
244C2×Q8—48no—
245C2×Q8—84no—
246C2×Q8—84no—
247C2×Q8—84no—
248C2×Q8—84no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—42no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4C4×D412yes—
308C22×C4C4×D412yes—
309C22×C4#12112yes—
310C22×C4#12112yes—
311C22×C4#12412yes—
312C22×C4#12412yes—
313C22×C4C2×D814yes—
314C22×C4C2×D812yes—
315C22×C4C2×SD1614yes—
316C22×C4C2×SD1612yes—
317C22×C4#13512yes—
318C22×C4#13512yes—
319C22×C4C23×C411yescharacteristic
320C22×C4C22×D411yescharacteristic
321C22×C4C22×Q811yescharacteristic
322C22×C4C2×#4111yescharacteristic
323C22×C4C2×#4111yescharacteristic
324C22×C4C2×#4111yescharacteristic
325C22×C4C2×#4111yescharacteristic
326C22×C4C2×#4111yescharacteristic
327C22×C4C2×#4111yescharacteristic
328C22×C4C2×#4111yescharacteristic
329C22×C4H(5,𝔽2)11yescharacteristic
330C22×C42−1+411yescharacteristic
331C22×C42−1+411yescharacteristic
332C22×C42−1+411yescharacteristic
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—24no—
343C2×C8—24no—
344C2×C8—24no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8C4×Q814yes—
354C4⋊C4—28no—
355C4⋊C4—28no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—42no—
378C4⋊C4—41no—
379C4⋊C4—42no—
380C4⋊C4—48no—
381C4⋊C4—48no—
382C4⋊C4—84no—
383C4⋊C4—88no—
384C42—42no—
385C42—42no—
386C42—42no—
387C42—42no—
388C42—84no—
389C42—84no—
390C42#11714yes—
391C42C4×D414yes—
392C42#12314yes—
393C42#12614yes—
394C23C2×C4×D411yescharacteristic
395C23C2×#12111yescharacteristic
396C23C2×#12411yescharacteristic
397C23#54611yescharacteristic
398C23#54711yescharacteristic
399C23#54811yescharacteristic
400C23D4×Q811yescharacteristic
401C23C22×D811yescharacteristic
402C23C22×SD1611yescharacteristic
403C23C2×#13511yescharacteristic
404C23C2×#13511yescharacteristic
405C23#57711yescharacteristic
406C23#57711yescharacteristic
407C23#57811yescharacteristic
408C23#57811yescharacteristic
409Q8—816no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—82no—
454C2×C4—81no—
455C2×C4—82no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—82no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—82no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4#38814yes—
473C2×C4#38814yes—
474C2×C4C4×D814yes—
475C2×C4C4×SD1614yes—
476C2×C4#46514yes—
477C2×C4#46614yes—
478C2×C4#48614yes—
479C2×C4#48814yes—
480C2×C4C2×#11711yescharacteristic
481C2×C4C2×C4×D411yescharacteristic
482C2×C4C2×C4×Q811yescharacteristic
483C2×C4C4×#4111yescharacteristic
484C2×C4#51911yescharacteristic
485C2×C4#52011yescharacteristic
486C2×C4C2×#12311yescharacteristic
487C2×C4C2×#12611yescharacteristic
488C2×C4#52911yescharacteristic
489C2×C4#52911yescharacteristic
490C2×C4#53311yescharacteristic
491C2×C4#53311yescharacteristic
492C2×C4#54111yescharacteristic
493C2×C4#54111yescharacteristic
494C2×C4#54211yescharacteristic
495C2×C4#54211yescharacteristic
496C2×C4#54611yescharacteristic
497C2×C4#54711yescharacteristic
498C2×C4#54811yescharacteristic
499C2×C4D4×Q811yescharacteristic
500C2×C4#55211yescharacteristic
501C2×C4#55211yescharacteristic
502C2×C4#55411yescharacteristic
503C2×C4#55611yescharacteristic
504C2×C4#55611yescharacteristic
505C2×C4#55611yescharacteristic
506C2×C4#55711yescharacteristic
507C2×C4#55911yescharacteristic
508C8—24no—
509C8—42no—
510C8—44no—
511C22C2×#38811yescharacteristic
512C22#236511yescharacteristic
513C22#236711yescharacteristic
514C22#242711yescharacteristic
515C22#242911yescharacteristic
516C22#248911yescharacteristic
517C22#249211yescharacteristic
518C22C2×C4×D811yescharacteristic
519C22C2×C4×SD1611yescharacteristic
520C22#294911yescharacteristic
521C22#295011yescharacteristic
522C22C2×#46511yescharacteristic
523C22C2×#46611yescharacteristic
524C22#306311yescharacteristic
525C22#306311yescharacteristic
526C22C2×#48611yescharacteristic
527C22C2×#48811yescharacteristic
528C22#313711yescharacteristic
529C22#313711yescharacteristic
530C22#329411yescharacteristic
531C22#329811yescharacteristic
532C22#330211yescharacteristic
533C22#330411yescharacteristic
534C22#333411yescharacteristic
535C22#333511yescharacteristic
536C22#334011yescharacteristic
537C22#334211yescharacteristic
538C22#335811yescharacteristic
539C22#335911yescharacteristic
540C22#336411yescharacteristic
541C22#336511yescharacteristic
542C22Q8×D811yescharacteristic
543C22Q8×SD1611yescharacteristic
544C22#338111yescharacteristic
545C22#338311yescharacteristic
546C4—24no—
547C4—44no—
548C4—44no—
549C4—82no—
550C4—84no—
551C4—82no—
552C4—82no—
553C4—84no—
554C4—82no—
555C4#193418yes—
556C4#229511yescharacteristic
557C4#232011yescharacteristic
558C4#236011yescharacteristic
559C4#236211yescharacteristic
560C4#244611yescharacteristic
561C4#244711yescharacteristic
562C4#251111yescharacteristic
563C4#251411yescharacteristic
564C2#1414211yescharacteristic
565C2C2×#193411yescharacteristic
566C2#2108811yescharacteristic
567C2#2315611yescharacteristic
568C2#2315711yescharacteristic
569C2#2321411yescharacteristic
570C2#2321611yescharacteristic
571C2#2533811yescharacteristic
572C2#2534111yescharacteristic
573C2#2546911yescharacteristic
574C2#2547211yescharacteristic
575C2#2772711yescharacteristic
576C2#2773011yescharacteristic
577C2#2781311yescharacteristic
578C2#2781611yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
24. #1990C4semidirect
111. #340Q8semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#19902
C2×#19882
C2×C8×C4⋊C42
#141552
#141512
#113152
#112922
#108802
#108492
#108202
C4×#3402